SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Measurement, STD2 M6 2023 HSC 12 MC

A cylindrical pipe with a radius of 12.5 cm is filled with water to a depth, `d` cm, as shown.

The surface of the water has a width of 20 cm.
 

What is the depth of water in the pipe?

  1. 2.5 cm
  2. 5.0 cm
  3. 7.5 cm
  4. 12.5 cm
Show Answers Only

`B`

Show Worked Solution

`text{Triangle base}\ = 20/2=10\ text{cm}`

`text{Let}\ x =\ text{⊥ distance from centre to water}`

`text{Using Pythagoras:}`

`12.5^2` `=x^2+10^2`
`x^2` `=12.5^2-10^2=56.25`
`x` `=\sqrt{56.25}=7.5\ text{cm}`

 
`d=12.5-7.5=5.0\ text{cm}`
  

`=>B`

♦ Mean mark 42%.

Measurement, STD2 M6 2019 HSC 22

Two right-angled triangles, `ABC` and `ADC`, are shown.
 

Calculate the size of angle `theta`, correct to the nearest minute.  (3 marks)

Show Answers Only

`41°4^{′}\ \ text{(nearest minute)}`

Show Worked Solution

`text(Using Pythagoras in)\ DeltaACD:`

Mean mark 51%.

`AC^2` `= 2.5^2 + 6^2`
  `= 42.25`
`:.AC` `= 6.5\ text(cm)`

 
`text(In)\ DeltaABC:`

`costheta` `= 4.9/6.5`
`theta` `= cos^(−1)\ 4.9/6.5`
  `= 41.075…`
  `= 41°4^{′}31^{″}`
  `= 41°5^{′}\ \ text{(nearest minute)}`

Measurement, STD2 M6 2016 HSC 26d

The diagram shows a block of land `ABCD` that has been surveyed. All measurements are in metres.
 

2ug-2016-hsc-q26_2
 

Calculate the length of `AB`, correct to the nearest metre.   (2 marks)

Show Answers Only

`46\ text{m}`

Show Worked Solution

2ug-2016-hsc-q26d-answer1

`text(Using Pythagoras,)`

`AB` `= sqrt(32^2 + 33^2)`
  `= 45.967\ …`
  `= 46\ text{m  (nearest m)}`

Measurement, STD2 M6 2006 HSC 26a

Daniel conducts an offset survey to sketch a diagram, `ABCD`, of a block of land.

Daniel walks from `A` to `C`, a distance of 62 m.

When he is 15 m from `A`, he notes that point `D` is 25 m to his right.

When he is 57 m from `A`, he notes that point `B` is 20 m to his left.

This is his notebook entry.

  1. Draw a neat sketch of the block of land. Label `A, B, C` and `D` on your diagram.   (1 mark)

  2. Calculate the distance from `C` to `D`. (Give your answer to the nearest metre.)   (2 marks)

Show Answers Only

 a.

b.    `text{53 m (nearest m)}`

Show Worked Solution
a.    

 

b.    `text(Using Pythagoras:)`

`CD^2` `= 47^2 + 25^2`
  `= 2834`
`:. CD` `= 53.235…`
  `= 53\ text{m  (nearest m)}`

Measurement, STD2 M6 2005 HSC 25b

2UG-2005-25b

  1. Use Pythagoras’ theorem to show that `ΔABC` is a right-angled triangle.   (1 mark)

  2. Calculate the size of `∠ABC` to the nearest minute.   (2 marks)

Show Answers Only

a.    `text(Proof)`

b.    `67°23^{′}`

Show Worked Solution

a.    `ΔABC\ text(is right-angled if)\ \ a^2 + b^2 = c^2`

`a^2 + b^2` `= 5^2 + 12^2`
  `= 169`
  `= 13^2`
  `= c^2\ …\ text(as required.)`

MARKER’S COMMENT: Know your calculator process for producing an angle in minutes/seconds. Note >30 “seconds” rounds up to the higher “minute”.

 
b.    `sin ∠ABC = 12/13`

`:.∠ABC` `= 67.38\ …°`
  `=67°22^{′}48^{″}`
  `= 67°23^{′}\ \ \ text{(nearest minute)}`

Measurement, STD2 M6 2011 HSC 9 MC

Two trees on level ground, 12 metres apart, are joined by a cable. It is attached 2 metres above the ground to one tree and 11 metres above the ground to the other.

What is the length of the cable between the two trees, correct to the nearest metre? 

  1. `9\ text(m)`
  2. `12\ text(m)`
  3. `15\ text(m)`
  4. `16\ text(m)`
Show Answers Only

`C`

Show Worked Solution

`text(Using Pythagoras)`

`c^2` `=12^2+9^2=144+81=225`
`c` `=\sqrt{225}=15,\ \ c>0`

 
`=>C`

Measurement, STD2 M6 2010 HSC 3 MC

A field diagram has been drawn from an offset survey.
 

What is the distance from `G` to `H` correct to the nearest metre?

  1. 11
  2. 13
  3. 16
  4. 20
Show Answers Only

`B`

Show Worked Solution

`text(Using Pythagoras:)`

`GH^2=12^2+(16-11)^2=144+25=169`

`GH=sqrt169=13\ text(m)`

` =>  B`